I may have been that guy...
One of the things causing me grief these days is the inconsistent ability to "do" aikido. I have broken down things to try to better understand where the "doing" part happens. In that respect, I have consolidated around the concept of "expressing/demonstrating aiki" and demonstrating a practical application of aiki, "demonstrating aikido." Essentially, I am just talking about technique done with aiki. But, I believe that aiki is real and been shown, which is not what everyone else thinks. I also believe many of us are not actually using aiki in our technique.
We can demonstrate aiki without framing that demonstration in a practical application (aikido). The most common demonstration I can think of is a push test. Or a pull test. The old guys did these a lot when they shared aikido with other arts. Pushing heads, jos, arms, etc. Mochizuki did these alot with judo players and there is a good clip of Sunadomari doing it in his friendship demo. Tohei, O Sensei, Shioda. They all did demonstrations that had these types of elements. I believe these sihan were showing aiki without the facade of technique to hide what they were doing.
The point that I was getting at was that we (aikido people) should be able to show someone "aiki" in a variety of different ways so spectators could see aikido without risk of injury. My unspoken comment was that maybe we don't understand aiki as well as we think if we only have a limited manner in which we can share what we do. Any jujutsu girl can show you an armbar and hurt you in the process - how do we separate ourselves if we cannot show aikido without hurting someone?
When we demonstrate aikido, we pick a collection of techniques in which we excel in expressing aiki and we present those techniques as an example of what aikido can do. But, I think sometimes we use kata to mask the lack of aiki. If I do good jujutsu kata, how is that dissimilar from good aikido kata? Hopefully, not much different. If I know someone will fall down from good jujutsu, who is to say my demonstration is not just good jujutsu? It has to be felt. Back to the practical idea of demonstrating aiki on someone with technique... How can we just pull someone out of the crowd to feel it if we are going to hurt them?
Oddly, I started thinking about this several years ago when our shihan, Saotome sensei, stopped teaching kata. Most of his seminars now are aiki "tricks." the centipede ukes, no touch throws, push throws... I think he felt that "kata" was getting in the way of "aiki". While I don't have the newest ASU handbook, I am pretty sure "centipede nage waza" is not part of the curriculum on which we test. I believe sensei is trying to show us the real motor that drives his techniques.
I think our collection of techniques were created for a common theme - the practical application of aiki. When you have the aikibody, the themes tend to constantly exist throughout movement. Is it different than modern aikido? Yes. But not everyone wants that. I also talked about aikido's tent size - there is a lot of variation in aikido and each one has some value to those who practice it.
I am also a student under Saotome sensei in ASU and based on my experiences with sensei and others whom I know and train with, your description of his teaching does not reflect at least what I have seen and heard from him at seminars as well as private teaching at the shrine. I have not noticed any substantial changes in his teachings over the past few years. I have seen him teaching waza with committed attacks from Uke and drawing upon techniques that depend utilizing the momentum from Uke's attack. While it is certainly accurate from my experience with him that he teaches other things that resemble exercises or slow motion technique.
However, when he does so I do not believe he is teaching what you refer to as "aiki" (the definition of aiki being internal strength). It is certainly possible for one to read into these exercises and see something else though. However, over the years Sensei has explained what most of them are for so there would be no need to guess what they mean. Typically, they address blending, being flexible and being willing to change. An example would be a static ikkyo that connects to the shoulder or the blending with very small amount of energy. Also, you mention the push tests from Ki Society as an example of people doing what you are defining as "aiki" training for the aiki body, but it is my understanding that Ki Society under Tohei believe their grounding ability results from the projection of Ki not IS. Not true?
Finally, sensei may do a few things that are about connecting in a way that you are defining as "aiki" but those are by far a minority of what he demonstrates in my experience and certainly IMO doesn't constitute a paradigm shift in his teaching. The majority of what Saotome sensei is describing in my experience is blending with Uke's energy to get kuzushi, to discombobulate.
"Aikido techniques depends on blending with the force of the attack. It is that force which determines the movement....." Page 180 Aikido and the Harmony of Nature
Again, its just my observation and the experience of those whom I train or interact with who train with Sensei too.